Invariants
| Level: | $45$ | $\SL_2$-level: | $45$ | Newform level: | $405$ | ||
| Index: | $648$ | $\PSL_2$-index: | $324$ | ||||
| Genus: | $22 = 1 + \frac{ 324 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 12 }{2}$ | ||||||
| Cusps: | $12$ (none of which are rational) | Cusp widths | $9^{6}\cdot45^{6}$ | Cusp orbits | $3^{2}\cdot6$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $2$ | ||||||
| $\Q$-gonality: | $4 \le \gamma \le 12$ | ||||||
| $\overline{\Q}$-gonality: | $4 \le \gamma \le 12$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 45A22 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 45.648.22.5 |
Level structure
| $\GL_2(\Z/45\Z)$-generators: | $\begin{bmatrix}3&8\\11&25\end{bmatrix}$, $\begin{bmatrix}8&1\\4&0\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 45.324.22.a.1 for the level structure with $-I$) |
| Cyclic 45-isogeny field degree: | $12$ |
| Cyclic 45-torsion field degree: | $144$ |
| Full 45-torsion field degree: | $2880$ |
Jacobian
| Conductor: | $3^{80}\cdot5^{22}$ |
| Simple: | no |
| Squarefree: | yes |
| Decomposition: | $1^{3}\cdot2^{2}\cdot3\cdot4^{3}$ |
| Newforms: | 135.2.a.c, 135.2.a.d, 135.2.b.a, 405.2.a.a, 405.2.a.d, 405.2.a.e, 405.2.a.j, 405.2.b.a, 405.2.b.d |
Rational points
This modular curve has no $\Q_p$ points for $p=7,13$, and therefore no rational points.
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 15.24.0-5.a.1.2 | $15$ | $27$ | $27$ | $0$ | $0$ | full Jacobian |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 45.1944.64-45.a.1.3 | $45$ | $3$ | $3$ | $64$ | $7$ | $1^{12}\cdot2^{2}\cdot3^{2}\cdot4^{5}$ |
| 45.2592.85-45.k.1.3 | $45$ | $4$ | $4$ | $85$ | $9$ | $1^{12}\cdot2^{7}\cdot3^{3}\cdot4^{7}$ |
| 45.3240.118-45.i.1.3 | $45$ | $5$ | $5$ | $118$ | $24$ | $1^{10}\cdot2^{13}\cdot3^{2}\cdot4^{10}\cdot6\cdot8$ |